https://file.aiquickdraw.com/mathbot/file/1726482619847onb11hz1.jpg

The problem in the image asks to find the limit:

\[
\lim_{{h \to 0}} \frac{\sqrt{20h + 4} - 2}{h}
\]

To solve this, we need to rationalize the numerator, which involves multiplying both the numerator and denominator by the conjugate of the numerator.

### Step 1: Multiply by the conjugate

The conjugate of \(\sqrt{20h + 4} - 2\) is \(\sqrt{20h + 4} + 2\). So, multiply both numerator and denominator by this conjugate:

\[
\frac{\sqrt{20h + 4} - 2}{h} \cdot \frac{\sqrt{20h + 4} + 2}{\sqrt{20h + 4} + 2} = \frac{(\sqrt{20h + 4} - 2)(\sqrt{20h + 4} + 2)}{h(\sqrt{20h + 4} + 2)}
\]

### Step 2: Simplify the numerator

The numerator simplifies using the difference of squares formula:

\[
(\sqrt{20h + 4} - 2)(\sqrt{20h + 4} + 2) = (\sqrt{20h + 4})^2 - 2^2 = (20h + 4) - 4 = 20h
\]

Now the expression becomes:

\[
\frac{20h}{h(\sqrt{20h + 4} + 2)}
\]

### Step 3: Cancel \(h\)

Since \(h\) is in both the numerator and denominator, we can cancel \(h\) (assuming \(h \neq 0\)):

\[
\frac{20}{\sqrt{20h + 4} + 2}
\]

### Step 4: Take the limit as \(h \to 0\)

Now, substitute \(h = 0\) into the expression:

\[
\lim_{{h \to 0}} \frac{20}{\sqrt{20(0) + 4} + 2} = \frac{20}{\sqrt{4} + 2} = \frac{20}{2 + 2} = \frac{20}{4} = 5
\]

Thus, the value of the limit is:

\[
\boxed{5}
\]

Would you like further details on this solution or additional clarification? Here are 5 related questions:

1. How is the difference of squares formula used in rationalizing the numerator?
2. Why do we multiply by the conjugate when simplifying limits involving square roots?
3. What other techniques can be used for solving limits that involve radicals?
4. How do we handle limits when the expression becomes undefined at \(h = 0\)?
5. Can we apply L'Hopital's Rule to this limit problem instead of rationalizing the numerator?

**Tip:** When simplifying limits involving radicals, always consider rationalizing the numerator or denominator to eliminate the square root.

Find the limit: lim (h -> 0) [sqrt(20h + 4) - 2] / h. Rationalize the numerator of the expression.

Learn how to solve the limit problem involving square roots by rationalizing the numerator. This problem demonstrates how to apply the difference of squares and limit laws. Suitable for students in college-level calculus.

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